The value of ${}^{n-1}{\text{C}}_0.{}^1{\text{C}}_0+{}^{n-1}{\text{C}}_1.{}^n{\text{C}}_2+{}^{n-1}{\text{C}}_2.{}^n{\text{C}}_3+\cdots +{}^{n-1}{\text{C}}_{n-1}.{}^n{\text{C}}_n$ is equal to 

${}^{n-1}{\text{C}}_0.{}^1{\text{C}}_0+{}^{n-1}{\text{C}}_1.{}^n{\text{C}}_2+{}^{n-1}{\text{C}}_2.{}^n{\text{C}}_3+\cdots +{}^{n-1}{\text{C}}_{n-1}.{}^n{\text{C}}_n$

Consider ${(1+x)}^n={}^n{\text{C}}_0+{}^n{\text{C}}_{1}x+{}^n{\text{C}}_2{x}^2+\cdots +{}^n{\text{C}}_n{x}^n$

${\left(1+\frac{{\displaystyle 1}}{{\displaystyle x}}\right)}^{n-1}={}^{n-1}{\text{C}}_0+{}^{n-1}{\text{C}}_1\left(\frac{{\displaystyle 1}}{{\displaystyle x}}\right)+{}^{n-1}{\text{C}}_2{\left(\frac{{\displaystyle 1}}{{\displaystyle x}}\right)}^2+\cdots +{}^{n-1}{\text{C}}_{n-1}{\left(\frac{{\displaystyle 1}}{{\displaystyle x}}\right)}^{n-1}$

$\begin{array}{l}{(1+x)}^n{\left(1+\frac{1}{x}\right)}^{n-1}=[{}^n{\text{C}}_0+{}^n{\text{C}}_{1}x+{}^n{\text{C}}_2{x}^2+\cdots +{}^n{\text{C}}_n{x}^n]\\ [{}^{n-1}{\text{C}}_0+{}^{n-1}{\text{C}}_1\left(\frac{1}{x}\right)+{}^{n-1}{\text{C}}_2{\left(\frac{1}{x}\right)}^2+\cdots +{}^{n-1}{\text{C}}_{n-1}{\left(\frac{1}{x}\right)}^{n-1}]\end{array}$

$\begin{array}{l}\frac{{(1+x)}^{2n-1}}{x^{n-1}}=[{}^n{\text{C}}_0+{}^n{\text{C}}_{1}x+\cdots +{}^n{\text{C}}_n{x}^n]\\ \left[{ }^{n-1}{\text{C}}_0+{}^{n-1}{\text{C}}_1\left(\frac{1}{x}\right)+\cdots +{}^{n-1}{\text{C}}_{n-1}{\left(\frac{1}{x}\right)}^{n-1}\right]\end{array}$

Equating x coefficient on both sides

$\frac{x^n\text{coefficient\hspace{0.17em}\hspace{0.17em}in\hspace{0.17em}\hspace{0.17em}}{(1+x)}^{2n-1}}{x^{n-1}}={}^{n-1}{\text{C}}_0{}^n{\text{C}}_1+{}^{n-1}{\text{C}}_1{}^n{\text{C}}_2+\cdots$

$\begin{array}{l}{}^{2n-1}{\text{C}}_n={}^{n-1}{\text{C}}_0.{}^n{\text{C}}_1+{}^{n-1}{\text{C}}_1{}^n{\text{C}}_2+\cdots \\ \therefore {}^{n-1}{\text{C}}_0{}^n{\text{C}}_1+{}^{n-1}{\text{C}}_1{}^n{\text{C}}_2+\cdots ={}^{2n-1}{\text{C}}_n\\ ={}^{2n-1}{\text{C}}_{2n-1-n}\\ ={}^{2n-1}{\text{C}}_{n-1}\end{array}$